Unselected vertices are moving along, and my proportional editing is off
Welcome to Blender Stack Exchange! You've come to the right place. The reason you're seeing that is due to a setting in the subdivision surface modifier. It's the little button with the triangle on it (the one made of 3 dots). That changes how you can interact with the modifier. Enabling it as you have places the vertices on the smoothed mesh, but can make it harder to control as you see. Disabling it will make the base mesh more like a cage, with the smoothed high resolution mesh inside.